You can ask students to find the number of degrees
in angle A of triangle ABC if angle B equals 72 degrees and angle C
is 34 degrees, and that is a reasonable and acceptable math
problem.

But if
you ask a student to find the number of degrees between the hands of
a clock at 4:08, then this is more interesting math
problem.

What if you pose the problem in this way: "Find the number of
degrees between the hands of a clock at 4:08. Explain how to do this
problem as if you were explaining it to someone who did not know how
to do the problem, and came to you for help. Include diagrams and a
complete explanation. Then find the number of degrees between the
hands of a clock at 4:32, and explain how to do this also. Are there
any differences in the method used for 4:08 and 4:32?"

Now you
have a challenging question, an interesting answer, and "writing in
mathematics"!

This
was actually the first writing assignment I had every assigned, and I
really didn't know what to expect, both in terms of the student's
reaction to the idea of writing in a math class, and is terms of what
they might actually write.

The
students had done four simple "clock problems" from the text book,
problems such as "Find the measure of the angle between the hands
of a clock at 3 PM". If you consider that question, you will
realize that the answer is simple, because the minute hand is
"straight up; (vertical) and the hour hand is directly on the numeral
3, aso they hands simply form a right angle. What complicates the
problem for times such as 4:08 and 4:32 is that the minute hand moves
as the hour hand moves, and the angle between them changes every
minute. A superficial glance at a clock might cause one to think that
the calculations for other times might be also simple, but they are
not, and it is a rather sophisticated math problem, one that the
students found challenging. They not only rose to the challenge, but
wrote excellent explanations, and presented their work in creative
and imaginative ways! As a teacher, I believe that it is extremely
important to challenge students in this way, and to give them
opportunities to rise to the challenge!

They
worked in groups of 3 or 4, and turned in either a "poster" or a
"booklet" with illustrations and explanations. The work that my
students did in answering this question was fascinating! They
included wonderful diagrams, and interesting explanations. Some
solved the problem in ways I had not anticipated, and all of them
learned from the experience. Some of their work on this writing
project is truly excellent,

In
their explanation, one group wrote a very detailed explanation,
complete with carefully drawn diagrams:

The
Clock Problem

By
Jenna, Reyn, and Elizabeth:

"The
basic principle starts with the fact that there are 360 degrees of
angular measurement in a circle. This is applied to the Clock Problem
because a non-digital clock is a circle."

"Next we had to determine the
measure in degrees between the increment of 1 hour. Since there are
12 hours on a standard clock face we divided the total 360 degree
measurement by 12 to get 30. 360/12=30. This tells us that the space
between any 2 hours is 30 degrees."

"A
clock's full measure is 360 degrees.
Therefore, between each pair of numbers (i.e. 10
and 11) it is 30 degrees. We figured this out because there are 12
pairs, and 360 divided by 12 = equals 30. At 4:08,the clock looks
something like this:

The students went on to give a
very detailed written explanation of the rest of the steps in solving
this problem. I don't know whether they drew the diagrams themselves,
or what computer software they used, but they certainly did an
excellent job. Other groups also drew diagrams as part of the
solution, but theirs were less "professional", but their written
explanations were equally good.

Here is another example of student work on this interesting
problem:

THE CLOCK PROBLEM

by Chris, Sam, Lance and
Jared:

"1) A clock face is divided
into 12 hours, and 60 minutes. There are 59 marks on the clock, with
60 spaces between them in total. They represent every minute in the
hour. There are also 12 hour marks that show the hours, represented
by numerals 1-12. The entire clock face is 360 degrees. 360 divided
by 60 is 6, so you know that there are 6 degrees between one minute
mark and the next. 2) At 4:08, the minute hand is 8 spaces between
minute marks away from the twelve. 6 times 8 is 48, so the minute
hand is making a 48 degree angle from the number
twelve."

"3.) The numeral four on the
clock is 20 minute mark spaces away from the 12 because there are
(60/112) 5 minute mark spaces in every space between the hour marks.
So it is making a (20x6) 120 degree angle from the
twelve.

4.) The hour hand also moves
between the hour marks as corresponding to the minute hand going
around the clock. The minute hand is 40/360 of the way around the
clock, or 2/15. Therefore, the hour hand is 2/15 of the way between
the numerals 4 and 5, making a 30 degree angle (5 x 6 = 30). 2/15
times 30 degrees is 4 degrees. Add that to the angle the four is from
twelve (120 degrees) and you get that the hour hand is 124 degrees
away from the twelve.

5) To find the number of
degrees between the two hands of the clock, just subtract their
angles from the number 12. The hour hand has a 124 degree angle, and
the minute hand has a 48 degree angle. 124 minus 48 is 76. The two
hands make a 76 degree angle; they are 76 degrees
apart.

To find the number of degrees
between the hands a on a clock at 4:32, we can use steps similar to
the previous steps above."

"1) There are 6 degrees in the
space between every pair of minute marks, and the minute hand is 32
minute mark spaces away from the 12. 32 times 6 is 192, so the minute
hand is making a 192 degree angle from the twelve.

2) The numeral makes a 120
degree angle from the 12, so we must also add in the movement between
the hour marks. The minute hand is 192/360 of the way around the
clock face, or 8/15. There are 30 degrees between the numerals four
and five, and 8/15 times 30 degrees is 16 degrees. 16 degrees added
to 120 degrees puts the hour hand at a 136 degree angle from the
twelve.

3) To find the distance between
the hands, we will subtract again. 192 minus 136 is 56. The hands
make a 56 degree angle between them. (They are 56 degrees
apart.)

There is no difference in the
two methods, except that in the later example we subtracted the
degrees of the hour hand from the degrees of the minute hand instead
of the other way around. I you wanted both methods to be the same,
you could just say that the distance between the two hands is the
absolute value of the difference of the angles made by the two hands
to the numeral 12 (going clockwise)."

I frequently asked the students to
write reflections on the process of solving a problem, and have
included some of their reflections below. "I liked this project
because I gained a lot from doing it. I also learned I could explain
things clearer when using diagrams. It made me feel real proud to see
what we accomplished." Jon R

Working
in groups was very helpful, in that they were not overwhelmed by the
task when they could get help from one another, and the interaction
was wonderful to listen to. It was great to hear them argue about the
math concepts, come to an agreement, and learn from each
other.

One
student, Rachel, actually created a formula for solving this problem!
Here is her explanation:

Then
the measure of the angle between the hands of the clock at 3:45 =

30 (H
+ M/60)
- 6M)

Shen
added the comment "if you get an angle more than 180 degrees,
subtract from 360. If you get a negative angle, don't worry about it,
just ignore the negative sign."

In
their reflections on this project, students made some very
interesting comments:

"I
think that the clock problem was important because it took teamwork.
At first I didn't understand what my group was doing, but when I
chipped in, I understood what the group and myself was thinking. I
had my own way of doing it, but I found out that there is more than
just one way of doing something." Shelton L.

In
every project, I asked them to write what they thought about the
project, which parts they found challenging, and what they learned in
doing it. They never ceased to amaze me with their work, and their
comments. Here is one student's response:

"This
assignment represents the biggest AHA moment of the course for me. At
first I thought I would never get it. Suddenly it hit me that the
angle increased 5 1/2 degrees per minute, and I was on my way to
answering the problem." David S.

In many
of the projects and other assignments in this writing-intensive
geometry class, I asked my students to write reflections on what they
experienced, and what they learned. I found the results of this
particular assignment so interesting that I wrote my own reflections
on it. This was actually the first writing assignment, and I really
didn't know what to expect, both in terms of their reaction to the
idea of writing, and in terms of what they would actually write.

They
had done four simple "clock problems", as a warm-up excercise -
finding the angle between the hands of a clock at